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Haptics and Virtual Reality

M. Zareinejad

Lecture 11:

Haptic Control

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fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world

although these approximation errors are small, their potentially non-passive nature can have profound effects, notably:

instability limit cycle oscillations (which can be just as

bad as instability)

why do instabilities occur?

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a useful tool for studying the stability and performance of haptic systems a one-port is passive if the integral of power extracted over time does not exceed the initial energy stored in the system.

Passivity

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Passive system properties

A passive system is stable

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“Z-width” is the dynamic range of impedances that can be rendered with a haptic display while maintaining passivity

we want a large z-width, in particular: zero impedance in free space large impedance during interactions with

highly massive/viscous/stiff objects

Z-width

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lower bound depends primarily on mechanical design (can be modified through control)

upper bound depends on sensor quantization, sampled data effects, time delay (in

teleoperators), and noise (can be modified through control)

in a different category are methods that seek to create a perceptual effect (e.g., event-based rendering)

How do you improve Z-width?

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sampled-data system example

stability of thevirtual wall

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Direct coupling – limitations in 1DOF No distinction between simulation and control:

◦ VE used to close the force feedback loop.

In 1DOF:◦ Energy leaks. ZOH. Asynchronous switching.

Passivity and stability A passive system is stable. Any interconnection of passive systems

(feedforward or feedback) is stable.

Real contact

◦ No contact oscillations.◦ Wall does not generate

energy (passive).

Haptic contact

◦ Contact oscillations possible.◦ Wall may generate energy (may

be active).

Haptic vs. physical interaction

Direct coupling – network model

• Impedance– Measures how much a

system impedes motion.– Input: velocity.– Output: force.

• Varies with frequency.

Terminology – interaction behavior

Interaction behavior = dynamic relation among port variables (effort & flow).

• Admittance– Measures how much a

system admits motion.– Input: force.– Output: velocity.

)()()(svsFsZ )(

)()()( 1 sZsFsvsY

Interaction behavior of ideal mass (inertia)

mssvsFsZ

sFsmsvFym L

)()()(

)()(

..

Interaction behavior of ideal spring

sK

svsFsZ

svs

KsF

KyF L

)()()(

)(1)(

Interaction behavior of ideal dashpot

BsvsFsZ

sBvsFyBF L

)()()(

)()(

Impedance behavior of typical mechanical system

sKBsms

svsFsZ

svsKBmssF

KyyBymF L

2

)()()(

)()(

19

20

21

22

23

24

25

26

27

28

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Direct coupling – network model (ctd. II)

Continuous time system:

Sampled data system – never unconditionally stable!

e

h

e

hd

e

h

fv

zzb

zz

Tm

fvzZOHzZ

vf

***01

121

112

01)()(

e

h

e

h

fvbms

vf

011

Virtual coupling – mechanical model

• Rigid

• Compliant connection between haptic device and virtual tool

Virtual coupling – closed loop model

Virtual coupling – network model

e

h

vc

d

e

h

vc

d

e

h

e

d

vce

d

fv

zZ

zZOHzZ

fv

zZ

zZOHzZvf

fv

zZvf

***

*

*

*

)(11

)()(

)(1110

01)()(

)(1110

where:

Tzzs

vcvcvc s

KBzZ1

)(

)()(Re2)(cos1

)(1Re

0)(

1Re

0))(Re(

zZOHzZzZOH

zZ

zZ

zZ

dvc

vc

d

Virtual coupling – absolute stability

• Llewellyn’s criterion:

• Need:• Physical damping.• “Give” during rigid contact.• Larger coupling damping for larger contact stiffness.• Worst-case for stability: loose grasp during contact.• Not transparent.

evc

eZOHvcdh ZZ

ZZZZZ

Virtual coupling – admittance device • Impedance / admittance duality:

• Spring inertia.• Damper damper.

• Need:• “Give” during contact.• Physical damping.• Larger coupling damping for larger contact stiffness.• Worst case for stability: rigid grasp during free motion.

Teleoperation Unilateral:

◦ Transmit operator command.

Bilateral:◦ Feel

environment response.

Ideal bilateral teleoperator• Position & force matching:

• Impedance matching:

• Intervening impedance:

• Position & force matching impedance matching.

sm

sm

ffxx

teth ZZ

s

m

s

m

fv

vf

0110

temth ZZZ

s

mm

s

m

fvZ

vf

011

Bilateral teleoperation architectures• Position/Position (P/P).• Position/Force (P/F).• Force/Force (F/F).• 4 channels (PF/PF).• Local force feedback.

General bilateral teleoperator

General bilateral teleoperator (ctd.)

ess

emmssmmth ZCCCCCZ

ZCCCZCCCZCZZ2343

2141

1

• Impedance transmitted to user:

• Perfect transparency:

• Trade-off between stability & transparency.

eth ZZ

Haptic teleoperation Teleoperation:

◦ Master robot.◦ Slave robot.◦ Communication

(force, velocity).

Haptics:◦ Haptic device.◦ Virtual tool.◦ Communication

(force, velocity).

4 channels teleoperation for haptics• Transparency requirement:

– Stiffness rendering (perfect transparency):

– Inertia rendering (intervening impedance):

• Control/VE design separated.

eth ZZ

esth ZZZ

Direct coupling as bilateral teleoperation• Perfect transparency.

• Potentially unstable.emth ZZZ

Virtual coupling as bilateral teleoperation

cm

esc

escmth

ZZZZZZZZZZ

• Not transparent.

• Unconditionally stable.

Stability analysis • Questions:• Is the system stable?• How stable is the system?

• Analysis methods:• Linear systems:

• Analysis including Zh and Ze:• Non-conservative.• Need user & virtual environment models.

• Analysis without Zh and Ze:• Zh and Ze restricted to passive operators.• Conservative.• No need for user & virtual environment models.

• Nonlinear systems – based on energy concepts (next lecture):• Lyapunov stability.• Passivity.

Stability analysis with Zh and Ze

• Can incorporate time delays.• Methods:

• Routh-Hurwitz:• Stability given as a function of multiple variables.

• Root locus:• Can analyze performance.• Stability given as function of single parameter.

• Nyquist stability.• Lyapunov stability:

• Stability margins not available.• Cannot incorporate time delay.

• Small gain theorem:• Conservative: considers only magnitude of OL.

• m-analysis:• Accounts for model uncertainties.

Stability analysis without Zh and Ze

• Absolute stability: network is stable for all possible passive terminations.• Llewellyn’s criterion:

• h11(s) and h22(s) have no poles in RHP.• Poles of h11(s) and h22(s) on imaginary axis are simple

with real & positive residues.• For all frequencies:

• Network stability parameter (equivalent to last 3 conditions):

• Perfect transparent system is marginally absolutely stable, i.e.,

1ReRe2cos2112

22112112

hhhhhh

0ReReRe20Re0Re

211221122211

22

11

hhhhhhhh

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Stability analysis without Zh and Ze (ctd. I)

• Passivity:• Raisbeck’s passivity criterion:

• h-parameters have no poles in RHP.• Poles of h-parameters on imaginary axis are simple with

residues satisfying:

• For all frequencies:

0ImImReReReRe40Re0Re

22112

222112211

22

11

hhhhhhhh

12*

2121122211

22

11

000

kkwithkkkkkk

Stability analysis without Zh and Ze (ctd. II)

• Passivity:• Scattering parameter S:

• As function of h-parameters:

• Perfect transparent system is marginally passive, i.e.,

)()(

)()(

)()()(

)()(

svsv

sFsF

sSsvsv

sFsF

e

h

e

h

e

h

e

h

1)( sS

1)()(1001

)(

IsHIsHsS

Passivity – absolute stability – potential instability[Haykin ’70]

)Re()Re(2 2211

2112

hh

hh

)Re()Re(2)Re()Re(

2211

2112

hhhh